How do I rotate an object with a custom button and panel using python scripting?
One way might be a modal operator which tracks mouse movements and applies it to pose bone rotation. The operator modal template (Goto Scripting View -> Templates -> Python -> Operator modal) should be a good starting point.

Here is a small example:

import bpy
from bpy.props import IntProperty, FloatProperty

class ModalOperator(bpy.types.Operator):
    """Rotate a bone with the mouse, example"""
    bl_idname = "object.modal_operator"
    bl_label = "Simple Modal Operator"

    first_mouse_x = IntProperty()
    first_value = FloatProperty()

    def modal(self, context, event):
        if event.type == 'MOUSEMOVE':
            delta = self.first_mouse_x - event.mouse_x
            self.pbone.rotation_euler.x = self.first_value + delta * 0.1

        elif event.type == 'LEFTMOUSE':
            return {'FINISHED'}

        return {'RUNNING_MODAL'}

    def invoke(self, context, event):
        self.first_mouse_x = event.mouse_x
        self.pbone = context.selected_pose_bones[0] # &lt;- put your bone here instead
        self.first_value = self.pbone.rotation_euler.x

        context.window_manager.modal_handler_add(self)
        return {'RUNNING_MODAL'}


def register():
    bpy.utils.register_class(ModalOperator)


def unregister():
    bpy.utils.unregister_class(ModalOperator)


if __name__ == "__main__":
    register()

    # test call
    bpy.ops.object.modal_operator('INVOKE_DEFAULT')


You can then add your operator to your panel